Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C7×#126 ⋊ C2 |
| Sylow | C7 ⋊ #2823 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C2×C7×#126 | C2 |
| prime radical | C22×C22⋊D7 | C22 |
| hypercenter | C2×#126 | D7 |
| hypocenter | C7 | #2823 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^7,7|a^2=e,d^2=f,b^a=bef,c^a=cf,c^b=cg,d^b=de,h^b=h^6,d^c=def110 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#2823, C2×#11740564, #13560341, … |
| maximal normal subgroup of | D7×#2823, C2×#11740564, #13560341, … |
| maximal quotient of | #13284551, #13284560, #13284562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 15330);